Classical Boundary Conditions Research Articles

Free vibrations and dynamic behavior of the three layered flexoelectric functionally graded microbeam under moving load

In this study, free vibrations and the dynamic responses of the three layered flexoelectric functionally graded microbeam under the effect of moving load are investigated. Using the first order shear deformation and the size-dependent piezoelectricity theories in the framework of Hamilton's principle, the governing equations of motion and the corresponding classical and non-classical boundary conditions are obtained. Next, the equations of motion were discretized and solved using the finite element method. In order to validate the governing equations and the solution method, the obtained results have been compared with the results of previous studies, which shows a good convergence between the current and previous results. The obtained results demonstrate the dynamic behavior of the microbeam is strongly influenced by the geometrical ratios, the flexoelectric layers thickness, the functionally graded power index, the flexoelectric coefficient and the load velocity. So that the escalation in the strain gradient from the augmentation of the flexoelectric layers' thickness leads to a corresponding rise in the electric potential function. Moreover, as the flexoelectric coefficient increases, the microbeam becomes more responsive to polarization, thereby generating a greater amount of electrical energy. In addition, changes in the velocity of the moving load significantly affect the dynamic behavior of the structure in such a way that with the increase in the load velocity, the dynamic deflection of the structure increases.

Classical echoes of quantum boundary conditions

Abstract We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit.

A novel solution for dynamic behaviors of multi-span bridge plates

The present paper draws a comprehensive analysis of the free vibration characteristics and discusses modal localization phenomena of multi-span bridge plate structures through an analytical method, numerical simulations, and experimental measurements. A novel analytical solution using the generalized superposition segment method (GSSM) is first proposed to investigate the transverse vibration of a multi-span bridge, which offers complete flexibility in describing arbitrarily classical boundary conditions. Furthermore, a new experimental setup achieves precisely simply supported boundary conditions, and scaled physical models of two- and three-span bridge plates are employed to validate the analytical solution. The effect of span length mismatch is studied by analyzing the resonant frequencies and mode shapes of a multi-span bridge plate with varying span lengths. The good consistency of results by theoretical analysis, numerical simulation, and experimental measurements indicate that the proposed analytical solution can solve the vibration problem of the multi-span bridge plate efficiently and reflect the real-world boundary condition. Finally, the two modal localization behaviors are observed in theoretical analysis and experimental measurement. Through analytical solutions, numerical simulations, and experimental measurements, this study enhances the understanding of the dynamic behavior of multi-span bridges. It provides a new theoretical benchmark for predicting vibrations of multi-span bridge systems and useful insights into the transient behavior of multi-span bridge plates, suggesting several fruitful avenues for future research, including exploring passive and active control systems and vibration suppression techniques.

General Theory for Damped Beams with Elastic Supports Subjected to a Moving Damped Sprung Mass

Previously, classical boundary conditions (i.e. simple, fixed and cantilevered) have been adopted for the beams under the moving loads in deriving the dynamic response. In reality, most beams cannot be simply classified as the ones with classical supports, but are elastically restrained. For theoretical completeness, a general theory will be developed in this note for the damped beams with elastic restraints modeled by vertical and rotational springs subjected to a moving damped sprung mass. Essential to the present theory is the solution of the transcendental equation for the frequency by the bi-section method. The solution obtained can also be applied to the classical boundary conditions (i.e. simple, fixed and cantilevered) under a moving sprung mass. The reliability of the present theory, along with the bi-section method for solving the frequency and mode shape, is validated by comparison with the solution obtained by the finite element method (FEM) for various damping ratios, mass ratios and running speeds of the sprung mass.

Well-posedness of a bulk-surface convective Cahn–Hilliard system with dynamic boundary conditions

We consider a general class of bulk-surface convective Cahn–Hilliard systems with dynamic boundary conditions. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn–Hilliard type allow for dynamic changes of the contact angle between the diffuse interface and the boundary, a convection-induced motion of the contact line as well as absorption of material by the boundary. The coupling conditions for bulk and surface quantities involve parameters K,L∈[0,∞]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K,L\\in [0,\\infty ]$$\\end{document}, whose choice declares whether these conditions are of Dirichlet, Robin or Neumann type. We first prove the existence of a weak solution to our model in the case K,L∈(0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K,L\\in (0,\\infty )$$\\end{document} by means of a Faedo–Galerkin approach. For all other cases, the existence of a weak solution is then shown by means of the asymptotic limits, where K and L are sent to zero or to infinity, respectively. Eventually, we establish higher regularity for the phase-fields, and we prove the uniqueness of weak solutions given that the mobility functions are constant.

On elastic deformations of cylindrical bodies under the influence of the gravitational field.

Large elastic deformations of gravitating cylindrical bodies play a significant role in everyday life. On the other hand, tiny such deformations are relevant for state-of-the art experiments, as they affect the physical properties of materials under consideration, impacting wave propagation. This is of key importance for a recently planned experiment to explore the influence of the gravitational field on entangled photons propagating in waveguides. The purpose of this work is to determine this influence. We use the methods of linear elasticity, including thermoelasticity, to determine the stresses and strains of the medium. For this, the symmetry of the cylinder allows us to solve the problem by using Mitchell's solutions of the equations satisfied by the Airy functions. The boundary conditions are implemented by an approximation of the Hertz contact method. We calculate the displacements, the stresses and strains for several classes of boundary conditions, and give explicit solutions for a number of physically motivated configurations. The influence of the resulting deformations on the planned GRAVITES experiment is determined. The results are relevant for fiber interferometry experiments sensitive to the effects of the gravitational field on photon propagation. Our calculations give stringent bounds on the environmental variables, which need to be controlled in such experiments.

The Modeling Method for Vibration Characteristics Analysis of Composite-Laminated Rotationally Stiffened Plate

The composite-laminated rotationally stiffened plate is widely applied in aviation, aerospace, ship, machinery, and other fields. For structural design and optimization, to investigate the vibration characteristics is important. In this paper, a modeling method of composite-laminated rotationally plate is established. The first-order shear deformation theory (FSDT) and the modified Fourier series are applied to construct the admissible displacement function of the stiffened plate-coupled systems. On this basis, the energy function of composite-laminated rotationally stiffened plate is established. Combined with the artificial virtual spring technology, the proposed theory could be used to analyze the vibration characteristics of composite-stiffened plate-coupled systems with various classical boundary conditions or arbitrary elastic boundary conditions. The Rayleigh–Ritz method is used to solve the energy function. Thus, the vibration characteristics of the composite-laminated rotationally stiffened plate are obtained and analyzed. The correctness of the theoretical analysis model was verified through modal experiments. On this basis, the effect of some important parameters on the vibration characteristics of stiffened plate structures is studied, such as the number, thickness, and width of the laminated stiffener, varying structural parameters, and different boundary conditions. This study can provide the theoretical basis for the vibration and noise reduction of such structures.

Free vibration of the piezoelectric laminated composite beams under the elastic foundation

Due to its self-adaptability and electric-mechanical interaction, the piezoelectric laminated composite structure has received increased attention in the engineering community, with the intricacy of the actual scenario, the structural stability based on numerous foundations has also been recognized. In this paper, a unified formulation which is based on classical laminated beam is presented to conduct free vibration analysis of piezoelectric laminated beams subjected to classical boundary conditions with the varied elastic foundations. The semi-analytical solutions of layered beams in various boundaries have been obtained based on the two-dimensional piezoelectric elastic theory and state space-differential quadrature method (SS-DQM), the solution of the general structural equation has been generated introducing the electrical boundary conditions and interlayer continuity conditions with this understanding. The different elastic foundations have been used to calculate the frequency equation of the piezoelectric laminated construction. The proposed finite element model is validated through the analysis of the piezoelectric laminated composite beam (PLCB) using the SS-DQM, and the present solutions are compared with those available in the literature to confirm their validity. A systematic parameter study for the piezoelectric laminated beams with various boundary conditions, piezoelectric parameters, lamina numbers and thickness ratios is also performed. The results of the PLCBs with various boundary constraints under the elastic foundation are presented and they may be served as a benchmark for researchers in this field.

On the Solvability of a Singular Time Fractional Parabolic Equation with Non Classical Boundary Conditions

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense.

Peridynamic correspondence model with strain gradient elasticity for microstructure dependent size effects

This study presents a peridynamic (PD) correspondence model with strain gradient elasticity (SGE) to capture size effect on strength of nano- and micro-scale structures. The classical elasticity theory lacking a length scale parameter does not account for the microstructure-dependent size effects. Strain Gradient (SG) elasticity is an extension of the classical elasticity with a length scale parameter(s) which can be linked to the microstructure of the material. Peridynamic theory introduces damage into the constitutive relations in a natural way by allowing for interactions of a material point within its horizon which is referred to as the internal length parameter. The correspondence model of the PD theory permits the incorporation of the constitutive relationship of SG theory. Therefore, the present approach combines the effects of PD and SGE length scale parameters on the stiffness and strength of the material. The resulting equations present two length parameters: the horizon of a material point in PD theory and the characteristic length in SGE theory. The PD equation of motion with SGE (PDSG) is free of spatial derivatives and allows for the imposition of classical and nonclassical boundary conditions. The PDSG is first applied to investigate the deformation response of a one-dimensional nanoscale film subjected to a quasi-static tensile load and a dynamic load through an initial constant strain. The static response is compared with the analytical solution for varying ratio of SG length parameter to PD horizon. The dynamic response is compared with a computational solution while satisfying the nonclassical boundary conditions. Subsequently, the PDSG is applied to study the deformation response of a two-dimensional nanoscale film subjected to a quasi-static axial load and a specified tangential end displacement. These two cases are considered under special displacement constraints to compare with the available analytical solutions. Subsequently, a two-dimensional nanoscale film with or without an internal crack is subjected to uniform tensile loading. Finally, crack propagation is simulated under incremental displacement loading. The results capture the analytical predictions and the expected increase in stiffness with increasing ratio of SG length parameter to PD horizon.

Dynamic stiffness formulations for exact modal and dynamic response analysis of three-dimensional acoustic cavities in cylindrical coordinates

Exact dynamic stiffness (DS) models for three-dimensional (3D) cavities in cylindrical coordinates under any classical acoustic boundary conditions (BCs) are proposed in this paper. The paper first derives the exact general solutions that satisfy the governing differential equations. Then, these general solutions are used to develop the DS matrices for 3D cavities in cylindrical coordinates by substituting them into the acoustic pressure and normal velocity BCs. Subsequently, the Wittrick–Williams algorithm, along with the analytical expressions of the j0 count for various shaped cavities, is applied to compute natural frequencies. Finally, reliable and efficient techniques for dynamic response solution are employed. The DSM provides benchmark solutions for various 3D cavities. The convergence, universality, and efficiency of the proposed method in modal and response analysis are illustrated by comparing the results obtained using the DSM and FEM. The calculation time when using the DSM is only 0.1% of that when using the commercial finite element method, while maintaining the same accuracy. This research not only provides an effective technique for analyzing the cylindrical acoustic pressure field in engineering, but also establishes a foundation for the analysis of structural–acoustic coupling in tubular structures.

Free Vibration of Annular Plates Constrained by Translational and/or Rotational Springs on Outer and Inner Edges

This paper presents comprehensive lists of natural frequencies for vibration of thin isotropic annular plates with outer and inner edges elastically constrained by translational and rotational springs. A method is extended from a previously presented Ritz approach to include effects of the springs. In numerical examples, sixteen models are considered to cover all general cases of locating translational and/or rotational springs on outer and/or inner edges in addition to classical boundary conditions of free and simple supported edges. After convergence study and comparison test in specific cases are made to validate solution accuracy, frequency parameters of the sixteen models are summarized and frequency variations with increasing spring stiffness are illustrated.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

On the Hinch–Kim dualism between singularity and Faxén operators in the hydromechanics of arbitrary bodies in Stokes flows

We generalize the multipole expansion and the structure of the Faxén operator in Stokes flows obtained for bodies with no-slip to generic boundary conditions, addressing the assumptions under which this generalization is conceivable. We show that a disturbance field generated by a body immersed in an ambient flow can be expressed, independently on the boundary conditions, as a multipole expansion, the coefficients of which are the moments of the volume forces. We find that the dualism between the operator giving the disturbance field of an nth order ambient flow and the nth order Faxén operator, referred to as the Hinch–Kim dualism, holds only if the boundary conditions satisfy a property that we call Boundary-Condition reciprocity (BC-reciprocity). If this property is fulfilled, the Faxén operators can be expressed in terms of the (m, n)th order geometrical moments of the volume forces (defined in the article). In addition, it is shown that in these cases, the hydromechanics of the fluid-body system is completely determined by the entire set of the Faxén operators. Finally, classical boundary conditions of hydrodynamic applications are investigated in light of this property: boundary conditions for rigid bodies, Newtonian drops at the mechanical equilibrium, porous bodies modeled by the Brinkman equations are BC-reciprocal, while deforming linear elastic bodies, deforming Newtonian drops, non-Newtonian drops, and porous bodies modeled by the Darcy equations do not have this property. For Navier-slip boundary conditions on a rigid body, we find the analytical expression for low order Faxén operators. By using these operators, the closed form expressions for the flow past a sphere with arbitrary slip length immersed in shear and quadratic flows are obtained.

Study on the Fundamental Frequency and Dynamic Mode of Traveling Wave Vibration of Rotating PJCS

The vibrations of rotating joined conical–conical shells with classical supported conditions have been studied extensively. As a matter of fact, in some cases, these classical boundary conditions cannot exactly model actual situations. Moreover, theoretical frameworks on them are still limited. This research aims to investigate the fundamental frequencies and dynamic mode shapes of the traveling wave of the rotating porous metal material joined conical–conical thin shells (PJCS) with elastic supports. By utilizing artificial spring technology, arbitrary elastic supported boundary conditions and classical boundary conditions are achieved efficiently. A new dynamic model has been formulated with the help of the first-order shear deformation theory (FSDT) and Hamilton’s principle. By employing the generalized differential quadrature (GDQ) method along with stress boundary conditions and generalized eigenvalues, various factors such as porosity, semi-vertex angles and stiffness are analyzed for their impact on the fundamental frequencies of forward wave (FW), backward wave (BW) and mode shapes. The presented results are validated through the convergence and comparison studies from literatures. The interesting and novel results indicate that the in-plane displacement constraints have the most significant impact on the critical speed, while the lateral displacement constraint has the least effect. The vibrations are more easily excited for the part with a larger half vertex angle. Rotating PJCS with Type 1 has the biggest critical rotating speed.

An asymptotic formulation for boundary value problems in a non-local elastic half-space

The primary purpose of this study is to develop an asymptotic formulation for boundary value problems in a non-local elastic half-space. For the sake of simplicity, the non-locality is limited to the vertical direction, which is represented by a one-dimensional exponential kernel, and the problem is formulated within the framework of Eringen’s theory. The proposed asymptotic approach is based on the assumption that the internal characteristic length is significantly smaller than a typical wavelength. This assumption allows for the development of an asymptotic formulation that expresses the considered boundary value problem in terms of local stresses. Additionally, the formulation includes explicit correction terms to the classical boundary conditions, which arise from the non-local effects. As an example application of the derived formulation, the Rayleigh surface waves in a plane strain problem are considered. Finally, numerical results are presented for certain specific values of elastic parameters to illustrate the effects of non-locality on the analyzed system.

Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions

<abstract><p>Many scholars have lately explored fractional-order boundary value issues with a variety of conditions, including classical, nonlocal, multipoint, periodic/anti-periodic, fractional-order, and integral boundary conditions. In this manuscript, the existence and uniqueness of solutions to sequential fractional differential inclusions via a novel set of nonlocal boundary conditions were investigated. The existence results were presented under a new class of nonlocal boundary conditions, Carathéodory functions, and Lipschitz mappings. Further, fixed-point techniques have been applied to study the existence of results under convex and non-convex multi-valued mappings. Ultimately, to support our findings, we analyzed an illustrative example.</p></abstract>

Slip of styrene–butadiene rubbers: The effects of pressure and viscous heating

During the processing of elastomeric compounds under high flow rates, significant pressure drops (106–108 Pa) are encountered. Under such conditions, the viscosity of these compounds is significantly affected by pressure and viscous heating. Moreover, strong flow rates may cause these systems to slip at the wall, violating the classical no-slip boundary condition of fluid mechanics. To determine the slip velocity by the well-known Mooney method, the effects of pressure and viscous heating should be considered. In this work, an experimental methodology is developed to determine the slip velocity of styrene–butadiene compounds in capillary flow corrected for the effects of pressure and viscous heating. First, the temperature increase due to viscous heating is measured during the extrusion process and accounted for in correcting the experimental data to infer the slip velocity. Consequently, the corrected experimental data for the effects of pressure and viscous heating are used to calculate the slip velocity from the deviation of the linear viscoelastic behavior (deviation from the Cox–Merz rule). The Mooney method is also used to confirm the calculated slip velocity of the elastomeric compounds.

On the well posedness of a mathematical model for a singular nonlinear fractional pseudo-hyperbolic system with nonlocal boundary conditions and frictional damping terms

<abstract><p>This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system with frictional damping terms. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedness of the nonlinear fractional system. This mathematical model of pseudo-hyperbolic systems arises mainly in the theory of longitudinal and lateral vibrations of elastic bars (beams), and in some special case it is propounded in unsteady helical flows between two infinite coaxial circular cylinders for some specific boundary conditions.</p></abstract>

Envelope enrichment method for homogenization of non-periodic structures

The homogenization of composite materials is critical for accurately predicting their mechanical performance, particularly when complex reinforcement arrangements are involved. Microstructural characteristics exert a substantial influence on the composite’s properties during practical applications. A widely adopted approach for analyzing composites is the numerical simulation of a Representative Volume Element (RVE). Although this method is well-established for periodic RVEs, it encounters difficulties when applied to non-periodic meshes, which complicates the imposition of classical boundary conditions on nodes exhibiting varying properties. To address this challenge, we propose a novel methodology that involves modeling an envelope surrounding the RVE, to which periodic boundary conditions are applied. By defining the envelope as a homogeneous material, stress transmission to the RVE is facilitated. The stiffness tensor of the envelope is updated iteratively through a homogenization process, ultimately converging to the effective properties of the RVE. The method is validated on a non-periodic arrangement of spherical inclusions embedded within a matrix. Convergence is observed in the different cases studied within ten iterations and the results are found within the Voigt and Reuss bound.